Abstract: The purpose of this study is to create and utilize a tool that evaluates students&apos; comprehension of the logical structure and implications of the formal definition of limit. This study continues the trajectory of recent limit research involving classroom-based interventions that reveal student metaphors and conception

Timothy Boester

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SIGMAA on RUME 2011 Proposal 
Contributed Research Report 
 
Designing and Implementing a Limit Diagnostic Tool 
 
Timothy Boester 
Wright State University 
 
 
 
Abstract: 
 
The purpose of this study is to create and utilize a tool that evaluates students’ 
comprehension of the logical structure and implications of the formal definition of limit. This 
study continues the trajectory of recent limit research involving classroom-based interventions 
that reveal student metaphors and conceptions (Boester, 2010; Oehrtman, 2009; Roh, 2008, 
2010). The diagnostic tool, based on seven concepts embedded in the formal definition, uses a 
set of delta/epsilon diagrams that students must explain, either accepting them as correct, or 
augmenting them to make them correct. The assessment was used after giving students in a 
conceptually-based calculus class a problem meant to introduce the logical structure of the 
formal definition. While students did not spontaneously show many of the concepts based on the 
problem alone, an interview protocol following the assessment prompted the students to rethink 
the implications of the problem, thus promoting the missing concepts. 
 
Keywords: calculus, limit, assessment, conceptual decomposition 
 
 
Cornu (1991) first summarized research on students’ spontaneous conceptions, mental 
models, and epistemological obstacles concerning the formal definition of limit. Since then, 
research has grown from cataloging misconceptions (Bezuidenhout, 2001; Davis & Vinner, 
1986; Tall & Vinner, 1981) to describing possible frameworks of student conceptions (Cottrill et 
al., 1996; Lakoff & Núñez, 2001; Williams, 1991, 2001). Some of the most recent research 
(Boester, 2010; Oehrtman, 2009; Roh, 2008, 2010) has described classroom-based interventions 
that both further our understanding of students’ conceptions of limit, while documenting how 
and why limits were taught to students using particular problems, activities, or manipulatives. 
In order to assess the effectiveness of emerging pedagogical strategies for limit 
instruction, it would be nice to have a generic tool to gauge students’ comprehension of the logic 
contained within the formal definition. The purpose of this study is to create such a tool, then use 
it to assess students’ comprehension of the logical structure and implications of the formal 
definition of limit following a classroom-based intervention. 
 
Limit Diagnostic Tool 
 
Using the following statement of the formal definition of limit at a point 
 
lim
x→a
f (x) = L  means that  
∀ε > 0 , ∃δ > 0 , such that 0 < x − a < δ ⇒ f (x) − L < ε  
 
the concepts contained within this definition, which should be assessed by the diagnostic tool, 
need to be established. Through discussion with other limit researchers, the following list was 
created: 
 
1) We control delta, not epsilon. 
2) Delta interval must fit inside (cannot be outside) epsilon interval. 
3) Delta interval can be strictly inside epsilon interval. 
4) Delta and epsilon do not have to be equal. 
5) In order for the limit as x approaches a to be L in the continuous case, f(a) = L. 
6) The length of the interval on each side of a / L must be the same, because you 
can be the same distance (delta / epsilon) away in both directions. 
7) For a non-linear graph, one side of the delta interval may be the same as the 
epsilon interval, but for the other side, the delta interval may be strictly inside 
the epsilon interval. 
 
A written assessment, consisting of six standard delta/epsilon diagrams (an example is 
shown in Figure 1), was then created to test for these seven concepts. For each diagram, the 
student must either confirm that this pairing of delta and epsilon in the diagram is appropriate for 
the given graph of a function, or explain why it is not appropriate and correct the diagram to 
create an appropriate pairing. For instance, if a student states that the diagram in Figure 1 is not 
an appropriate pairing, and proposes redrawing the delta interval to “correct” the diagram in 
Figure 2, this would reveal that the student has failed to grasp concept #3, that the delta interval 
can be strictly inside the epsilon interval. A student could alternatively redraw the epsilon 
interval to line up with the delta interval, showing that they have also failed to grasp concept #1, 
that we can only change delta. 
In addition to the written assessment, an interview protocol was also created. After a few 
preliminary questions eliciting feedback about limits in general, students are asked to explain 
their responses to all six diagrams. The interviewer questions their responses only after the 
student has completed explaining all the diagrams, in order to provide a baseline for their 
responses. For correct responses, the questions probe for recognition of the embedded concepts. 
Did the student actually answer the question following the intent of the concept? In other words, 
were the students’ beliefs robust enough to withstand deeper questioning, or were they easily 
malleable by the researcher? For example, if a student correctly states that the diagram in Figure 
1 follows the definition, the protocol indicates to show the student Figure 2 and ask “Another 
student might say that the delta is too small, and should be bigger to match the epsilon. Do you 
need to do this?” For partially correct or incorrect responses, the questions probed for the related 
concepts in their apparent absence. Would students recognize or maintain their misconceptions 
of the definition? If a student proposed Figure 2 as a correction to Figure 1, the student would be 
asked “Is it ok for the delta interval to be inside of the epsilon interval? Do delta and epsilon 
have to be equal?” (This addresses concept #4, as well as concept #3.) 
 
Implementation of the Tool Following a Classroom Activity 
 
This diagnostic tool was first used in Math 348, Concepts of Calculus for Middle School 
(Pre-Service) Teachers, a course taught by the researcher during the Spring 2010 quarter at a 
mid-size, Midwestern university. Math 348 focuses on the ideas, rather than the procedures, of 
calculus. Even though the goal of the course is to enable students to recognize how the 
fundamental idea of change relates to the functions commonly presented in middle school 
curricula, limits are introduced in the course, mainly to promote derivatives and integrals later 
on. 
Limits were first covered informally, based on a dynamic, approaching conception. 
Students were asked to solve routine limit problems for continuous, discontinuous, and 
piecewise-defined functions. Then the formal definition was introduced through a classroom 
activity centered around a story problem originally created for a teaching experiment (Boester, 
2010). The bolt manufacturing problem allows students to explore the logical structure of the 
formal definition by thinking about the functional relationship between the input and output of a 
factory that makes bolts. After allowing students to discuss the bolt problem in groups, a whole 
class discussion (led by the researcher) was held to come up with the following statement: 
 
For every bolt length tolerance, there exists a raw materials tolerance, so that if an 
amount of raw material that falls within the raw material tolerance is put into the 
machine, the length of the bolt produced falls within the bolt length tolerance. 
 
Written on the board next was a delta/epsilon diagram, whose graph had a slope roughly equal to 
one so that the delta and epsilon intervals would match and be equal. (The researcher tried to 
match the diagrams drawn by the students during their small group discussions of the problem.) 
This diagram was initially labeled with the terms from the bolt problem. The formal definition of 
limit was then written on the board. Finally, the researcher walked the students though a 
mapping of each representation onto the other, showing in particular how the symbols of the 
definition matched the pieces of the statement and the diagram. 
The concepts on the above list were intentionally not highlighted during instruction. 
While all of these concepts are embedded in the bolt problem, could students actually unpack 
them without further explicit instruction? The students were reminded, however, to think of the 
bolt problem while constructing their answers and explanations for each diagram.  
Students were then given the assessment tool as a written, optional homework assignment 
for extra credit. (Students did not have to agree to participate in the study to complete the 
assignment and receive the extra credit.) Those students who completed the assessment were also 
given the opportunity to be interviewed. Out of 24 students enrolled in the course, 13 returned 
the written survey, and 8 of those sat for a videotaped interview. For the students who were 
interviewed, their remarks were transcribed and compared to their written solutions in two 
passes, once for their initial explanations, and a second time when their responses were probed 
by the researcher. For those students that were not interviewed but turned in the assessment, their 
written remarks were coded as if they had sat for the first pass of the interview.  
 
Results 
 
A preliminary analysis of the results has revealed that the students struggled with the 
assessment. Most students showed that they grasped concept #2, that the delta interval cannot be 
outside the epsilon interval. Interviewed students attributed their explanations of concept #2 to 
the bolt problem, that an amount of raw materials too far away from the target amount would 
produce a bolt with a length outside the acceptable length range. Students also showed that they 
grasped concept #4, that the delta and epsilon intervals do not have to be equal. One student 
explained that delta and epsilon are describing different qualities, so why should they have to be 
equal? 
However, few students expressed knowledge of concepts #1, #3 and #5. Evidence of this 
was shown through changing the epsilon interval to correct a diagram, expanding a delta interval 
to match an epsilon interval (as in Figure 2), and not recognizing that a misalignment of a and L 
in a diagram was a fundamental error, even if the delta interval was still within the epsilon 
interval. The lack of these concepts seemed to hinder their grasp of concepts later on the list (and 
tested in later diagrams in the assessment). While the last two errors could be attributed to 
metaphorical “baggage” of the bolt problem (in the context of the problem, these are acceptable 
or even desired actions), students should have recognized that the bolt manufacturer cannot 
control the customers’ demands on the accuracy of the bolt length (epsilon), only their own 
measurement error of the raw material (delta). 
Even with this disappointing performance, the second pass of the interview showed 
entirely different results. When probed, although a few students backed away from correct 
concepts, every student gained several concepts. The students’ responses to the probes were 
consistently framed in the context of the bolt problem, which clearly helped the students come to 
grasp the concepts they heretofore lacked. Even with the researcher being careful to closely 
adhere to the interview protocol, six of the eight students left the interviews having expressed 
every concept on the list. Thus, while the bolt problem itself may not be sufficient to instill the 
logical structure and implications of the formal definition of limit, the problem clearly primed 
students for a discussion of the diagrams used in the diagnostic tool. This suggests that an 
instructional sequence involving the bolt problem should not be considered complete without an 
assignment like that of the limit diagnostic tool, and a discussion of the assignment that questions 
the reasoning of correct and incorrect responses. 
References 
 
Bezuidenhout, J. (2001). Limits and continuity: some conceptions of first-year students. International 
Journal of Mathematical Education in Science and Technology, 32, 487–500. 
Boester, T. (2010). Testing Conceptual Frameworks of Limit: A Classroom-Based Case Study. 
Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics 
Education. 
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Dordrecht, 
The Netherlands: Kluwer Academic. 
Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). 
Understanding the limit concept: Beginning with a coordinated process scheme. Journal of 
Mathematical Behavior, 15, 167–192. 
Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception 
stages. Journal of Mathematical Behavior, 5, 281–303. 
Lakoff, G. & Núñez, R. (2001). Where mathematics comes from: How the embodied mind brings 
mathematics into being. New York: Basic Books. 
Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit 
concepts. Journal for Research in Mathematics Education, 40, 396–426. 
Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. 
Educational Studies in Mathematics, 69, 217–233. 
Roh, K. H. (2010). An empirical study of students’ understanding of a logical structure in the definition of 
limit via the ε-strip activity. Educational Studies in Mathematics, 73, 263–279. 
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular 
reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. 
Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in 
Mathematics Education, 22, 219–236. 
Williams, S. R. (2001) Predications of the limit concept: An application of reperatory grids. Journal for 
Research in Mathematics Education, 32, 341–367. 
 
 
 
 
Figure 1. The delta/epsilon diagram from 
Question #2. 
 
 
Figure 2. A proposed student correction to the 
delta/epsilon diagram in Question #2. 
 


Designing and Implementing a Limit Diagnostic Tool

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